Palindromic Prime Embedded Trees - RGMIA

RGMIA Research Report Collection, 20(2017), pp.1-14, http://rgmia.org/v20.php

Inder J. Taneja

Palindromic Prime Embedded Trees Received 10/08/17

Inder J. Taneja1 Abstract The idea of embedding palindromic prime (palprime) numbers in the form of tree is very famous in the literature, where previous palprime is in the middle of next, and so on. In this case there is no limit where it ends, because always we find next palprime containing previous one. In this work, we brought embedded palprimes trees starting with 3 digits.

Contents 1 Introduction

1

2 Comparison Study 2.1 Embedded Palprime Patterns . . . . 2.2 Fixed Digits Prime Patterns . . . . . 2.3 Symmetric Embedded . . . . . . . . . 2.4 Complimentary Embedded Palprimes 2.5 Magic-Square-Type Palprimes . . . .

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2 2 3 3 3 4

3 Palprime Embedded Trees 3.1 Palindromic Prime Embedded Trees 3.2 Embedded Tree with 101 . . . . . . 3.2.1 Embedded Tree with 131 . 3.2.2 Embedded Tree with 151 . 3.2.3 Embedded Tree with 181 . 3.2.4 Embedded Tree with 191 . 3.2.5 Embedded Tree with 313 . 3.2.6 Embedded Tree with 353 . 3.2.7 Embedded Tree with 373 . 3.2.8 Embedded Tree with 383 . 3.2.9 Embedded Tree with 727 . 3.2.10 Embedded Tree with 757 . 3.2.11 Embedded Tree with 787 . 3.2.12 Embedded Tree with 797 . 3.2.13 Embedded Tree with 919 . 3.2.14 Embedded Tree with 929 .

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5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13

1

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Introduction

Embedded palindromic prime numbers are very much famous in the literature [1, 2]. It is generally known by pyramid palprimes. Since, the previous palprime is in the middle of next one, we call it embedded palprimes. The work here is concentrated on all the palprimes of digits 3 and 5. We can always find a next palprime containing previous, we limit our study to length 6 for the 3 digits and length 5 for digits 5. By length here we understand that total number of primes in each pattern. In each case we can always find next. 1

Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florian´opolis, SC, Brazil. E-mail: [email protected]; Web-site: http://inderjtaneja.com; Twitter: @IJTANEJA.

RGMIA Res. Rep. Coll. 20 (2017), Art. 124, 14 pp 1


Inder J. Taneja

There are only 4 primes of single digit, i.e., 2, 3, 5 and 7. There is only one palprime of two digits, i.e., 11. There are total 15 palprimes of digits 3 given by 101

131

373 383

151 727

181 757

191 787

313 797

353 919

929.

101

101

131

131

151

191

31013

91019

11311

71317

31513

71917

313

353

373

383

727

757

93139

33533

93739

13831

37273

37573

757

787

797

797

929

97579

97879

17971

77977

39293

Thus, we work with 15 palprimes of digit 3 and 76 of 5 digits. There are much more possibilities of palprimes that we have given below. The aim is not to given more results, but cover all the possible palprimes of digits 3 and 5. We considered all the possible digits, i.e., 0 to 9. Next our aim is to work with restricted digits, such as, 1,3; 1, 6, 9; 1, 2, 5; etc. In some cases complimentary embedded palprimes are also given. By complimentary, we under stand that if we change one digit with other still it remains palprime pyramid. A general study of embedded palprimes of 3 and 5 digits can be seen in author’s recent work [16].

2 2.1

Comparison Study Embedded Palprime Patterns

Let us consider the following embedded palprime pattern up to length 10: I 101 31013 3310133 933101339 1593310133951 13159331013395131 171315933101339513171 1617131593310133951317161 96161713159331013395131716169 36396161713159331013395131716169363 ... (1) More study on this embedded palprimes can be seen in [1, 2, 3, 4].

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Inder J. Taneja

2.2

Fixed Digits Prime Patterns

Let us consider following pattern of prime numbers with fixed digit’s repetition: I 562 9933 562 192 9933 562 192 192 9933 562 192 192 192 9933 562 192 192 192 192 9933 562 192 192 192 192 192 9933 562 192 192 192 192 192 192 9933 562 192 192 192 192 192 192 192 9933 562 192 192 192 192 192 192 192 192 9933 562 192 192 192 192 192 192 192 192 192 9933. ... (2) The example (1) is of prime pattern with fixed digits repetition. The example (2) is embedded palprimes pattern, where previous palprimes is in the next. In case of example (1), we have digit 218 repeating horizontally as well as vertically. In total we have 10 prime numbers, while repetition is only in the nine prime numbers. In this case the next number is not a prime numbers, see below: 5621921921921921921921921921921929933 = 911 × 14264443 × 432625002551214522741563321. This means the process is limited. While in case of example (2), we can easily have further palprimes containing previous ones, see below: 323639616171315933101339513171616936323 772323639616171315933101339513171616936323277 ... ...

......

This means that we can always find next palprime embedding previous one, while the in case of prime pattern (1) , the process is limited. For more study on prime patterns refer author’s work [8, 9, 10, 11, 12].

2.3

Symmetric Embedded

Consider the following examples, 619

619

111619111

161619191

666111619111999

11616191911

6666611161911199999

61111616191911119

116666666111619111999999911

6666116161919119999

111661116666666111619111999999911199111

11666611616191911999911

6611166111666666611161911199999991119911199.

611111166661161619191199991111119. ... (3)

The above group of prime numbers are having the property of embedding primes, but are not palindromic. Moreover, they are symmetric in 6 and 9, but not changeable, i.e., if we replace 6 by 9 and 9 by 6, they are no more prime numbers.

2.4

Complimentary Embedded Palprimes

Below are examples of complimentary embedded palprimes. 3


Inder J. Taneja

16661

19991

1191166611911

1161199911611

1111111191166611911111111

1111111161199911611111111 ... (4)

131

191

71317

71917

77771317777

77771917777

11111777771317777711111 111111111111177777131777771111111111111

111111111111177777191777771111111111111

11111777771917777711111

... (5) 131

191

71317

71917

77771317777

77771917777

11777777771317777777711

11777777771917777777711

11111111111111777777771317777777711111111111111

11111111111111777777771917777777711111111111111

... (6)

In examples given in (4), the digits 6 and 9 are changeable. In examples given in (5) and (6), the digits 3 and 7 are changeable. Due this we call these types of examples as complimentary or paired embedded palprimes. Detailed study of this kind of results shall be dealt elsewhere.

2.5

Magic-Square-Type Palprimes

Let us consider the following two examples of magic-square-type palprimes of order 9 × 9 1 9 3 1 9 1 3 9 1

3 7 3 1 7 1 3 7 3

9 9 1 7 3 7 1 9 9

9 0 1 6 0 6 1 0 9

7 6 1 9 6 9 1 6 7

9 1 1 8 6 8 1 1 9

3 1 8 1 8 1 8 1 3

3 1 9 9 0 9 9 1 3

1 1 8 6 8 6 8 1 1

1 6 1 5 3 5 1 6 1

1 9 9 5 1 5 9 9 1

7 8 6 8 4 8 6 8 7

9 0 8 3 6 3 8 0 9

7 6 0 1 1 1 0 6 7

3 6 8 4 1 4 8 6 3

1 6 1 5 3 5 1 6 1

1 9 9 5 1 5 9 9 1

7 8 6 8 4 8 6 8 7

3 1 8 1 8 1 8 1 3

3 1 9 9 0 9 9 1 3

1 1 8 6 8 6 8 1 1

9 0 1 6 0 6 1 0 9

7 6 1 9 6 9 1 6 7

9 1 1 8 6 8 1 1 9

1 9 3 1 9 1 3 9 1

3 7 3 1 7 1 3 7 3

9 9 1 7 3 7 1 9 9 ... (6)

The above three examples are of symmetric-magic-square-type palprimes, i.e., all the numbers are palprimes in rows, columns and in principal diagonals. Moreover they have the property of row-wise embedding. See below: 908363809 161535161908363809161535161 318181813161535161908363809161535161318181813 901606109318181813161535161908363809161535161318181813901606109 193191391901606109318181813161535161908363809161535161318181813901606109193191391

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Inder J. Taneja

760111067 199515991760111067199515991 319909913199515991760111067199515991319909913 761969167319909913199515991760111067199515991319909913761969167 373171373761969167319909913199515991760111067199515991319909913761969167373171373 368414863 786848687368414863786848687 118686811786848687368414863786848687118686811 911868119118686811786848687368414863786848687118686811911868119 991737199911868119118686811786848687368414863786848687118686811911868119991737199 ... (7) All the three examples appearing in (7) have the property of embedding and also are palprimes. For detailed study refer author’s work [13, 14, 15].

3

Palprime Embedded Trees

Let us consider a 3 digits palprime 131. It is embedded in two palprimes 11311 and 71317. Again, 11311 is embedded in 121131121 and 141131141. Further 121131121 is embedded in 1212113112121 and 9312113112139. And 141131141 embedded in 11411311411 and 71411311417. Now the second one i.e., 71317 is embedded in 127131721 and 78271317287. Again 127131721 is embedded in 1712713172171 and 1812713172181. And 78271317287 is embedded in 1782713172871 and 3782713172873, and so on. In other words: 131 → 11311 → 121131121 → 1212113112121 → . . . 131 → 11311 → 141131141 → 11411311411 → . . . ...

...

...

See below the above structures in terms of tree: 1212113112121 121131121 9312113112139 11311 11411311411 141131141 71411311417 131 1712713172171 127131721 1812713172181 71317 1782713172871 78271317287 3782713172873 ... (8)

5


Inder J. Taneja

Alternatively, we can write the above structure in terms of 11311 and 71317 as embedded palprimes:

131

131

131

131

11311

11311

11311

11311

121131121

121131121

141131141

141131141

1212113112121

9312113112139

11411311411

71411311417

...

...

...

...

...

...

...

...

131

131

131

131

71317

71317

71317

71317

127131721

127131721

78271317287

78271317287

1712713172171

1812713172181

1782713172871

3782713172873

...

...

...

...

...

...

...

...

Remark 3.1. By no way, we can say that the above two-by-two example is unique. We may have other palprimes giving more splitting results. Also, this is not the end. This process continues as long as we go on finding more palprimes.

3.1

Palindromic Prime Embedded Trees

We have only 11 palprimes of with 3 digits. Below are palprime embedded trees starting with 3-digits. The example (8) given above goes up to 4th order and ending in 8 palprimes. The results given below goes up to 5th order and ends in 16 palprimes. We have splitted in two-by-two system, but there are much more possibilities.

3.2

Embedded Tree with 101 39993691019639993 9936910196399 33993691019639933 369101963 18963691019636981 9636910196369 15963691019636951 91019 38181091019018183 1810910190181 12181091019018121 109101901 30111091019011103 1110910190111 18111091019011181 101 933573101375339 35731013753 193573101375391 7310137 701773101377107 17731013771 121773101377121 31013 181533101335181 15331013351 1153310133511 3310133 135933101339531 933101339 105933101339501

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Inder J. Taneja

3.2.1

Embedded Tree with 131

731617131716137 3161713171613 131617131716131 16171317161 17116171317161171 1161713171611 11116171317161111 71317 12171271317217121 1712713172171 117127131721711 127131721 18151271317215181 1512713172151 15151271317215151 131 337141131141733 71411311417 9714113114179 141131141 711141131141117 11411311411 181141131141181 11311 95961211311216959 9612113112169 36961211311216963 121131121 99121211311212199 1212113112121 36121211311212163

3.2.2


1419381215121839141 193812151218391 7119381215121839117 38121512183 3615381215121835163 391212151212193 1515381215121835151 1215121 9239121215121219329 391212151212193 1239121215121219321 12121512121 1712121215121212171 121212151212121 91212121512121219 151 33187231513278133 1872315132781 15187231513278151 723151327 369723151327963 97231513279 189723151327981 31513 331233151332133 12331513321 111233151332111 3315133 761133151331167 11331513311 191133151331191

7


Inder J. Taneja

3.2.3


1710383918193830171 103839181938301 9210383918193830129 38391819383 17138391819383171 1383918193831 313839181938313 3918193 7975143918193415797 751439181934157 3175143918193415713 14391819341 1315143918193415131 151439181934151 9115143918193415119 181 19327321812372391 3273218123723 132732181237231 732181237 773732181237377 37321812373 353732181237353 3218123 12573321812337521 73321812337 307332181233703 332181233 343332181233343 33321812333 103332181233301

3.2.4


39327781918772393 3277819187723 10327781918772301 778191877 16117781918771161 1177819187711 111778191877111 7819187 70123781918732107 1237819187321 33123781918732133 378191873 133378191873331 33781918733 9337819187339 191 367747191747763 77471917477 3774719174773 747191747 763747191747367 37471917473 3374719174733 71917 13781471917418731 7814719174187 378147191741873 147191741 76391471917419367 3914719174193 139147191741931

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Inder J. Taneja

3.2.5


3916741831381476193 167418313814761 7216741831381476127 74183138147 70374183138147307 3741831381473 36374183138147363 1831381 7297351831381537927 973518313815379 39735183138153793 35183138153 73335183138153337 333518313815333 33335183138153333 313 323999313999323 39993139993 123999313999321 3931393 303299313992303 32993139923 193299313992391 93139 351839313938153 18393139381 111839313938111 3931393 121139313931121 11393139311 1113931393111

3.2.6


9994111035301114999 941110353011149 39411103530111493 11103530111 3430111035301110343 301110353011103 13011103530111031 1035301 33369103530196333 3691035301963 12369103530196321 910353019 71159103530195117 1591035301951 17159103530195171 353 99117333533371199 1173335333711 37117333533371173 733353337 363733353337363 37333533373 333733353337333 33533 361713353317163 17133533171 7171335331717 1335331 741213353312147 12133533121 351213353312153

9


Inder J. Taneja

3.2.7


3332761437341672333 327614373416723 13276143734167231 76143734167 78976143734167987 9761437341679 15976143734167951 1437341 74733143734133747 7331437341337 11733143734133711 314373413 12153143734135121 1531437341351 715314373413517 373 7533435937395343357 3435937395343 96343593739534369 359373953 18183593739538181 1835937395381 318359373953813 93739 3637712937392177363 7712937392177 10771293739217701 129373921 39321293739212393 3212937392123 34321293739212343

3.2.8


701101293383392101107 101293383392101 393101293383392101393 12933833921 36112933833921163 1129338339211 17112933833921171 9338339 1003219338339123001 3219338339123 72321933833912327 193383391 759193383391957 91933833919 309193383391903 383 16873213831237861 73213831237 13573213831237531 321383123 781321383123187 13213831231 121321383123121 13831 759141383141957 91413831419 709141383141907 141383141 723141383141327 31413831413 7314138314137

10


Inder J. Taneja

3.2.9


7318111572751118137 181115727511181 9918111572751118199 11157275111 17711157275111771 7111572751117 15711157275111751 1572751 10773157275137701 7731572751377 377315727513773 315727513 77363157275136377 3631572751363 30363157275136303 727 35157537273575153 1575372735751 18157537273575181 753727357 10297537273579201 97537273579 999753727357999 37273 78323337273332387 3233372733323 732333727333237 333727333 779333727333977 93337273339 309333727333903

3.2.10


78383697579638387 3836975796383 76383697579638367 369757963 77153697579635177 1536975796351 32153697579635123 97579 911159757951119 1115975795111 111159757951111 159757951 903159757951309 31597579513 333159757951333 757 907173757371709 71737573717 397173757371793 173757371 391173757371193 11737573711 331173757371133 37573 1021209375739021201 1209375739021 77120937573902177 9375739 701493757394107 14937573941 161493757394161

11


Inder J. Taneja

3.2.11


729911787119927 99117871199 309911787119903 911787119 351911787119153 19117871191 3191178711913 1178711 7547171178711717457 7171178711717 7217171178711717127 711787117 17367117871176371 3671178711763 11367117871176311 787 91381897879818319 3818978798183 138189787981831 189787981 189189787981981 91897879819 159189787981951 97879 153099787990351 30997879903 3309978799033 9978799 181799787997181 17997879971 1179978799711

3.2.12


73901177977110937 9011779771109 31901177977110913 117797711 32341177977114323 3411779771143 18341177977114381 77977 321897797798123 18977977981 121897797798121 9779779 901797797797109 17977977971 391797797797193 797 14751817971815741 7518179718157 11751817971815711 181797181 3331218179718121333 1218179718121 912181797181219 17971 331491797194133 14917971941 1149179719411 9179719 341191797191143 11917971911 191191797191191

12


Inder J. Taneja

3.2.13


922111635919536111229 111635919536111 351111635919536111153 16359195361 92916359195361929 9163591953619 71916359195361917 3591953 3421093591953901243 1093591953901 11109359195390111 935919539 121935919539121 19359195391 3193591953913 919 15930329192303951 9303291923039 12930329192303921 30329192303 19330329192303391 3303291923033 933032919230339 3291923 37153329192335173 1533291923351 13153329192335131 332919233 793332919233397 33329192333 303332919233303

3.2.14


7478361092901638747 783610929016387 97836109290163879 36109290163 1538361092901638351 383610929016383 73836109290163837 1092901 1134111092901114311 341110929011143 93411109290111439 11109290111 33311109290111333 3111092901113 731110929011137 929 11411739293711411 11739293711 391173929371193 7392937 3837392937383 373929373 3237392937323 39293 373333929333373 33339293333 103333929333301 3392933 7173392933717 733929337 3573392933753

13

Inder J. Taneja


Acknowledgement The author is thankful to T.J. Eckman, Georgia, USA (email: [email protected]) in programming the script to develop these representations.

References [1] G. L. HONAKER, JR. and CHRIS K. CALDWELL, Palindromic Prime Pyramids, http://www.utm.edu/staff/caldwell/. Also refer https://oeis.org/A053600 [2] G. L. HONAKER, JR. and CHRIS K. CALDWELL, Supplement to ”Palindromic Prime Pyramids”, http://www.utm.edu/staff/caldwell/supplements/. [3] P. De GEEST, Palindromic Prime Pyramid Puzzle, http://www.worldofnumbers.com/palprim3.htm. [4] CHRIS K. CALDWELL, The Prime Glossory, http://primes.utm.edu/glossary/xpage/PalindromicPrime.html. [5] I.J. TANEJA, Crazy Representations of Natural Numbers, Selfie Numbers, Fibonacci Sequence, and Selfie Fractions, RGMIA Research Report Collection, 19(2016), Article 179, pp.1-60, http://rgmia.org/papers/v19/v19a179.pdf https://goo.gl/cG0jdL . [6] I.J. TANEJA, 2017 - Mathematical Style, RGMIA, Research Report Collection, 20(2017), Article 03, pp.1-24, http://rgmia.org/papers/v20/v20a03.pdf - https://goo.gl/dKbbxU. [7] I.J. TANEJA, Hardy-Ramanujan Number - 1729, RGMIA Research Report Collection, 20(2017), Article 06, pp.1-50, http://rgmia.org/papers/v20/v20a06.pdf https://goo.gl/3LNf35. [8] I.J. TANEJA, Patterns in Prime Numbers: Fixed Digits Repetitions, RGMIA Research Report Collection, 20(2017), Article 17, pp.1-75, http://rgmia.org/papers/v20/v20a17.pdf - https://goo.gl/PquvOe. [9] I.J. TANEJA, Multiple Choice Patterns in Prime Numbers - I, RGMIA Research Report Collection, 20(2017), Art. 73, pp. 1-104, http://rgmia.org/papers/v20/v20a73.pdf - https://goo.gl/rPyzjr. [10] I.J. TANEJA, Multiple Choice Patterns in Prime Numbers - II, RGMIA Research Report Collection, 20(2017), Art. 74, pp. 1-109, http://rgmia.org/papers/v20/v20a74.pdf - https://goo.gl/1FwzLc. [11] I.J. TANEJA, Multiple Choice Patterns in Prime Numbers - III, RGMIA Research Report Collection, 20(2017), Art. 93, pp. 1-113, http://rgmia.org/papers/v20/v20a93.pdf - https://goo.gl/oW9EB6. [12] I.J. TANEJA, Multiple Choice Patterns in Prime Numbers - IV, RGMIA Research Report Collection, 20(2017), Art. 94, pp. 1-150, http://rgmia.org/papers/v20/v20a94.pdf - https://goo.gl/WbgsJE. [13] I.J. TANEJA, Magic Square Type Extended Row Palprimes of Orders 5x5 and 7x7, ÂăResearch Report Collection, 20(2017), Art. 21, pp. 1-69, http://rgmia.org/papers/v20/v20a21.pdf - https://goo.gl/Vv1v3G. [14] I.J. TANEJA, Magic Square Type Symmetric and Embedded Palprimes of Order 9x9 - I, Research Report Collection, 20(2017), Art. 22, pp. 1-63, http://rgmia.org/papers/v20/v20a22.pdf - https://goo.gl/62syas. [15] I.J. TANEJA, Magic Square Type Symmetric and Embedded Palprimes of Order 9x9 - II, Research Report Collection, 20(2017), Art. 23, pp. 1-92, http://rgmia.org/papers/v20/v20a23.pdf - https://goo.gl/9tsBH0. [16] I.J. TANEJA, Embedded Palindromic Prime Numbers - I, Research Report Collection, 20(2017), 1-86, http://rgmia.org/v20.php. ————————————————-

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